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October 26th, 2007, 08:54 PM   #1
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Conditional Expectation and Variance Question

Here's the question I'm having trouble with:
Show Var[E[Y|X]] = 0 given random variables X and Y are independent,
keeping in mind Var[Y] = Var[E[Y|X]] + E[Var[Y|X]].

Here's the work I've done so far.

1) Var[E[Y|X]] = E[(E[Y|X])^2] - (E[E[Y|X]])^2
(using Var[X] = E[X^2] - (E[X])^2)
2) Var[E[Y|X]] = E[E[Y|X]^2] - (E[Y])^2
(using E[Y|X] = E[Y] if X, Y are independent)
3) E[Y]^2 - E[Y]^2
(I want to use E[Y] = E[E[Y|X]] and but I need E[E[Y|X]^2] = E[E[Y|X]]^2

What am I missing here?
Thanks in advance.
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October 30th, 2007, 12:05 PM   #2
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Hmm.. That's a silly question your prof gave. I mean E(Y|X) = E(Y) due to independence, and so E(Y) is just a single number, of course Var(E(Y)) is 0...
Did I understand the question wrong?
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